CLASSIFICATION OF MOUFANG LOOPS OF ODD ORDER
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Date
2010-06
Authors
LOON, CHEE WING
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Abstract
The Moufang identity (x · y) · (z · x) = [x · (y · z)] · x was first introduced by
Ruth Moufang in 1935. Now, a loop that satisfies the Moufang identity is called
a Moufang loop. Our interest is to study the question: "For a positive integer n,
must every Moufang loop of order n be associative?". If not, can we construct a
nonassociative Moufang loop of order n?
These questions have been studied by handling Moufang loops of even and
odd order separately. For even order, Chein (1974) constructed a class of nonassociative
Moufang loop, M(G, 2) of order 2m where G is a nonabelian group of
order m. Following that, Chein and Rajah (2000) have proved that all Moufang
loops of order 2m are associative if and only if all groups of order m are abelian.
As for the case of Moufang loops of odd order, the existence of nonassociative
Moufang loops of order 34 and p5 (for any prime p > 3), has been shown by Bol
(1937) and Wright (1965) respectively. The most recent class of nonassociative
Moufang loops was constructed by Rajah (2001), where he showed that for distinct
odd primes p and q, there exists a nonassociative Moufang loop of order pq3
if and only if q 1 (mod p). On the other hand, the proofs on nonexistence of
nonassociative Moufang loops have progressed gradually for about four decades.
All Moufang loops of the following (odd) orders are known to be groups:
(
.) 0: !31 /32 (3 h dd . l p ql q2 ... qn n w ere p, ql) q2' ... ) qn are 0 pnmes, p < ql < q2 < ... <
qn, a ::;; 4 and f3i S: 2 (p > 3 if a = 4);
( ii) p1p2 · · · Pn q3 where P1 , P2, ... , Pn and q are distinct odd primes, q ¥= 1
(mod p1 ) and q2 ¥;. 1 (mod Pi) for all i E {2, ... , n}.
In this dissertation, we begin by defining the concept of minimally nonassociative
Moufang loops and proving some of their properties. From there, we
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continue with some of the known open cases for Moufang loops of particular odd
orders. We prove that Moufang loops of the following orders are groups:
(i) P1 · · · Pm q3r1 · · · r n where P1, ... , Pm, q, r1, ... , r n are odd primes, PI < · · · <
Pm < q < r1 < · · · < rn and q =/= 1 (mod Pi) for all i E {1,2, ... ,m};
( "") 2 2 3 2 2 h dd . 11 p1 · · · Pmq r 1 · · · rn w ere p1 , ... ,pm, q, r 1 , ... , rn are o pnmes, PI < · · · <
Pm < q < r1 < · · · < rn and q =/= 1 (mod Pi) for all i E {1, 2, ... , m};
(iii) p3q3 where p and q are odd primes, p < q and q :f= 1 (mod p); and
(iv) pq4 where p and q are odd primes, p < q and q :f= 1 (mod p).
In view of the fact that all the Moufang loops listed above are associative,
we turn our attention to the study of nonassociative Moufang loops of order 34
.
The classification done by Nagy and Vojtechovsky (2007) on these Moufang loops
was computer-aided. Hence, we give a theoretical proof of this result, establish
a product rule for any two elements in that Moufang loop and complete the
classification.
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CLASSIFICATION OF MOUFANG LOOPS OF ODD ORDER